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Spin-dependent tunneling into an empty lateral quantum dot

Motivated by the recent experiments of Amasha {\it et al.} [Phys. Rev. B {\bf 78}, 041306(R) (2008)], we investigate single electron tunneling into an empty quantum dot in presence of a magnetic field. We numerically calculate the tunneling rate from a laterally confined, few-channel external lead into the lowest orbital state of a spin-orbit coupled quantum dot. We find two mechanisms leading to a spin-dependent tunneling rate. The first originates from different electronic $g$-factors in the lead and in the dot, and favors the tunneling into the spin ground (excited) state when the $g$-factor magnitude is larger (smaller) in the lead. The second is triggered by spin-orbit interactions via the opening of off-diagonal spin-tunneling channels. It systematically favors the spin excited state. For physical parameters corresponding to lateral GaAs/AlGaAs heterostructures and the experimentally reported tunneling rates, both mechanisms lead to a discrepancy of $\sim$10% in the spin up vs spin down tunneling rates. We conjecture that the significantly larger discrepancy observed experimentally originates from the enhancement of the $g$-factor in laterally confined lead.